The tensor algebra of power series spaces
Volume 193 / 2009
Studia Mathematica 193 (2009), 189-202 MSC: Primary 46H20; Secondary 46A45, 46A63. DOI: 10.4064/sm193-2-5
The linear isomorphism type of the tensor algebra $T(E)$ of Fréchet spaces and, in particular, of power series spaces is studied. While for nuclear power series spaces of infinite type it is always $s$, the situation for finite type power series spaces is more complicated. The linear isomorphism $T(s)\cong s$ can be used to define a multiplication on $s$ which makes it a Fréchet m-algebra $s_\bullet$. This may be used to give an algebra analogue to the structure theory of $s$, that is, characterize Fréchet m-algebras with ($\Omega$) as quotient algebras of $s_\bullet$ and Fréchet m-algebras with (DN) and ($\Omega$) as quotient algebras of $s_\bullet$ with respect to a complemented ideal.